Parametric down-conversion from a wave-equations approach ...

Parametric down-conversion from a wave-equations approach: geometry and absolute brightness. Morgan W. Mitchell1 1

ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain

arXiv:0807.3533v3 [quant-ph] 25 Jan 2009

(Dated: 26 September 2008) Using the approach of coupled wave equations, we consider spontaneous parametric downconversion (SPDC) in the narrow-band regime and its relationship to classical nonlinear processes such as sum-frequency generation. We find simple expressions in terms of mode overlap integrals for the absolute pair production rate into single spatial modes, and simple relationships between the efficiencies of the classical and quantum processes. The results, obtained with Green function techniques, are not specific to any geometry or nonlinear crystal. The theory is applied to both degenerate and non-degenerate SPDC. We also find a time-domain expression for the correlation function between filtered signal and idler fields. I.

INTRODUCTION

Spontaneous parametric down-conversion (SPDC) has become a workhorse technique for generation of photon pairs and related states in quantum optics. Improvements in both nonlinear materials [1] and downconversion geometries have led to a steady growth in the brightness of these sources [2, 3, 4, 5, 6, 7, 8, 9]. Applications of the bright sources include fundamental tests of quantum mechanics, quantum communications, quantum information processing, and quantum metrology [10, 11, 12, 13]. Although down-conversion sources typically have bandwidths of order 1011 Hz, for the brightest sources even the output in a few-MHz window can be useful for experiments. This permits a new application, the interaction of down-conversion pairs with atoms, ions, or molecules. Indeed, sources for this purpose have been demonstrated [14]. Many modern applications use single-spatial-mode collection, either for improved spatial coherence, to take advantage of fiberbased technologies, or to separate the source and target for experimental convenience. Remarkably, despite the importance of bright, singlespatial-mode sources, general methods for calculating the absolute brightness of such a source are not found in the literature. By absolute brightness, we mean the number of pairs per second that are collected, for specified beam shapes, pump power, filters and crystal characteristics. A number of calculations study the dependence of brightness on parameters such as beam widths or collection angles, but these typically give only relative brightness: the final results contain an unknown multiplicative constant [15, 16]. While useful for optimizing a given source, they are less helpful when designing new sources. A recent paper computes the absolute brightness for a specific geometry: gaussian beams in the thin-crystal limit [17]. In this paper, we calculate the absolute brightness for

narrow-band, paraxial sources. The results are quite general, for example they apply equally well to crystals with spatial or temporal walk-off, for non-gaussian beams, etc. The Green-function approach we use is well suited to describing the temporal features of the down-conversion pairs, and we are able to predict the time correlations in a particularly simple way. To our knowledge, this method of deriving the time-correlations is also novel. Perhaps of greatest practical importance, we derive very simple relationships between the efficiency of classical parametric processes and their corresponding quantum parametric processes. For example, in any given geometry the efficiency of sum-frequency generation and spontaneous parametric down-conversion are proportional. This allows the use of existing classical calculations and/or experiments with classical nonlinear optics to predict the brightness of quantum sources. The paper is organized as follows: In section II, we describe briefly the variety of theoretical treatments that have been applied to parametric down-conversion, and our reasons for making a new calculation. In Section III we describe the formalism we use, based on an abstract paraxial wave equation and Green function solutions. In Section IV we calculate the absolute brightness and efficiencies for non-degenerate and degenerate parametric down-conversion and the corresponding classical processes. In section VI we summarize the results.

II.

BACKGROUND

The characteristics of parametric down-conversion light have been calculated in a number of different ways. Kleinman [18] used a Hamiltonian of the form H′ = −

1 3

Z

d3 x E · χ : E

(1)

2 and the Fermi “golden rule” to derive emission rates as a function of frequency and angle. Zel’dovich and Klyshko [19] proposed to use a mode expansion and calculate pair rates treating the quantum process as a classical parametric amplifier seeded by vacuum noise . Detailed treatment along these lines is given in [20, 21]. The problem of collection into defined spatial modes was not considered, indeed the works emphasize that the total rate of emission is independent of pump focusing. After the observation of SPDC temporal correlations by Burnham and Weinberg [22], Mollow [23] described detectable field correlation functions (coincidence distributions) in terms of source-current correlations and Green functions of the wave equation. This Heisenberg-picture calculation derived absolute brightness for multi-mode collection, e.g., for detectors of defined area at defined positions. It did not give brightness for single-mode collection, nor a connection to classical nonlinear processes. Hong and Mandel [24] used a mode-expansion to compute correlation functions based on the Heisenbergpicture evolution and an interaction Hamiltonian of the form Z 1 (2) HI = d3 x χijk Ei Ej Ek . (2) 2 As with Mollow’s calculation, they find singles and pair detection rates, but only for multi-mode detection [30] . Ghosh, et al. [25] used the same Hamiltonian in a Schr¨odinger-picture description, truncating the time evolution at first order to derive a “two-photon wavefunction.” This last method has become the most popular description of SPDC, including work on efficient collection into single spatial modes[15, 16]. Many works along these lines are cited in reference [15]. Recently, Ling, et al. [17] calculated the absolute emission rate based on a similar interaction Hamiltonian and a gaussian-beam mode expansion. In this way, they are able to calculate the absolute pair rate for non-degenerate SPDC in a uniform, thin crystal into gaussian collection modes. As described in Section V C, our calculation agrees with that of Ling et al. while also treating other crystal geometries, general beam shapes, and degenerate SPDC. Notable differences among the calculations include Heisenberg vs. Schr¨odinger picture and calculating in direct space vs. inverse space via a mode expansion. While they are of course equivalent, Heisenberg picture calculations are easier to compare to classical optics, while Schr¨odinger picture calculations are more similar to the state representations in quantum information. As our goal is in part to connect classical and quantum efficiencies, we use the Heisenberg picture. Also, we note that the Schr¨odinger-picture “two-photon wave-function” has a particular pathology: the first-order treatment of time evolution means the Schr¨odinger picture state is not normalized and never contains more than two downconversion photons. While this is not a problem for calculation of relative brightness or pair distributions[26, 27],

it does prevent calculation of absolute brightness. The choice of inverse vs. real space calculation is also one of convenience: for large angles in birefringent media or detection in momentum space, plane waves are the “natural” basis for the calculation. However, most bright sources use paraxial geometries and collection into defined spatial modes, e.g., the gaussian modes of optical fibers. In these situations, the advantages of a mode expansion disappear, while the local nature of the χ(2) interaction makes real-space more “natural.” Thus we opt for a real-space calculation. Our treatment of SPDC is based on coupled wave equations, a standard approach for multi-wave mixing in non-linear optics [28]. The calculations are done in the Heisenberg picture, so that the evolution of the quantum fields is exactly parallel to that of the classical fields described by nonlinear optics. This allows the re-use of well-known classical calculations such as those by Boyd and Kleinman [29]. As in the approach of Mollow, we use Green functions to describe the propagation, and find results that are not specific to any particular crystal or beam geometry. Unlike Mollow’s calculation, we work with a paraxial wave equation (PWE). This allows us to simply relate the classical and quantum processes through momentum-reversal, which takes the form of complex conjugation in the PWE. We focus on narrow-band parametric down-conversion, for which the results are particularly simple. By narrowband, we mean that the bandwidths of the pump and of the collected light are much less than the bandwidth of the SPDC process, as set by the phase-matching conditions. This includes recent experiments with very narrow filters [14], but also a common configuration in SPDC, in which the down-conversion bandwidth is ∼ 10 nm while the filter bandwidths are < 1 nm.

III.

A.

FORMALISM

description of propagation

We are interested in the envelopes E± for forwardand backward-directed of parts of the quantum field E (+) (t, x) = (E+ exp[+ikz] + E− exp[−ikz]) exp[−iωt] where k is the average wave-number and ω is the carrier frequency. These propagate according to a paraxial wave equation D± E± = S± ,

(3)

where D± is a differential operator and S± is a source term (later due to a χ(2) non-linearity).

3 The formal (retarded) solution to equation (3) is Z E± (x) = E0± (x) + d4 x′ G± (x; x′ )S± (x′ )

(4)

where x is the four-vector (t, x), E0± (x) is a solution to the source-free (S = 0) equation, and G± are the timeforward Green functions, defined by D± G± (x; x′ ) = δ 4 (x − x′ ) G± (x; x′ ) = 0 t < t′ .

(5)

For illustration, we consider the paraxial wave equation (PWE), for which D± ≡ ∇2T ± 2ik(∂z ± vg−1 ∂t )

where βt = 2ik/vg . Similar relationships hold for the advanced Green functions. If the field is known in some plane z = z0 downstream, then Z E± (x) = βz∗ d4 x′ H± (x; x′ )E± (x′ )δ(z ′ − z0 ) Z ∗ = βz∗ d4 x′ E± (x′ )δ(z ′ − z0 )G± (x′ ; x) (11) while if the field is known at some time tf in the future, Z ∗ E± (x) = βt d4 x′ H± (x; x′ )E± (x′ )δ(t′ − tf ) Z ∗ = βt∗ d4 x′ E± (x′ )δ(t′ − tf )G± (x′ ; x) (12)

(6)

2

S± =

ω (NL) P . c2 ε 0 ±

(7)

Here ∇2T is the transverse Laplacian, k = n(ω)ω/c is the wave-number, vg ≡ ∂ω/∂kz is the group velocity, and P (NL) is the envelope for the nonlinear polarization. We note that D± is invariant under translations of x, and that time reversal t → −t is equivalent to direction∗ reversal and complex conjugation, i.e., D± → D∓ . The results we obtain will be valid for any equation obeying these symmetries. In particular, the results will also apply to propagation with dispersion and/or spatial walkoff, which can be included by adding other time and/or spatial derivatives to D. From the symmetries of D± , it follows that the Green functions depend only on the difference x − x′ , and that ∗ G+ (t, x; t′ , x′ ) = G− (t, x′ , t′ , x). Also, the time-backward (or “advanced”) Green functions H± , defined by D± H± (x; x′ ) = δ 4 (x − x′ ) H± (x; x′ ) = 0 t > t′

(8)

∗ obey H± (x; x′ ) = G± (x′ , x).

B.

boundary and initial value problems

C.

quantization

The field envelopes are operators which obey the equaltime commutation relation [E(x, t), E † (x′ , t)] = A2γ δ 3 (x′ − x)

(13)

p where Aγ ≡ ~ω/2nng ε0 is a photon units scaling factor and ng ≡ c/v

g† is  the group index. For narrowE E describes a photon number denband fields, A−2 γ E D

†  −2 −2 † † sity, and vg Aγ E E and vgs vgi A−2 A E E E E γs s i i s deγi scribe single and pair fluxes. We find the unequal-time commutation relation from equation (10) [E(x), E † (x′ )] t>t′ = βt A2γ G(x; x′ )

(14)

so that h0| E(x)E † (x′ ) |0i = βt A2γ G(x; x′ ) for t > t′ . For 2 2 2 the PWE, A−2 γ vg = 2ncε0 /~ω and βt Aγ = i~ω /c ε0 . To calculate singles rates,

 we will need to evaluate expressions of the form EE † . For this, a useful expression is derived in the Appendix: Equation (A2) Z

 2~nω 3 d4 x′′ δ(z ′′ − z0 ) E(x)E † (x′ ) = c3 ε 0 ×G ∗ (x′′ ; x)G(x′′ ; x′ ).

(15)

Here z0 is any plane down-stream of x and x′ . If the value of the field is known on a plane z = zsrc , the field downstream of that plane is Z E± (x) = βz d4 x′ G± (x; x′ )E± (x′ )δ(z ′ − zsrc ) (9) where βz ≡ ±2ik. Similarly, if the field is known at an initial time t = t0 , the field later is Z E± (x) = βt d4 x′ G± (x; x′ )E± (x′ )δ(t′ − t0 ) (10)

D.

single spatial modes

A single spatial mode M± (x) is a time-independent solution to the source-free wave equation D± M± (x) = 0. ∗ M± (x) is the corresponding momentum-reversed solu∗ tion = 0. We assume the normalization R 3 D∓ M± (x) d x|M± (x)|2 δ(z) = 1. For single-mode collection, it

4 will be convenient to define the projection of a field E(x) onto the mode M as Z EM (t) ≡ d3 xM ∗ (x)δ(z − z0 )E(x) (16) (here and below, the +/− propagation direction is the same for E, M ). Here z0 is some plane of interest, and EM (t) describes the magnitude of the field component in this plane. Similarly, if the envelope is constant, the field distribution is E(x) = EM (t)M (x). The Roptical power is (MKS units) PM (t) 2ncε0 d3 x|E(t, x)|2 δ(z − z0 ) = 2ncε0 |EM (t)|2 .

(17) =

Given an upstream source S(x), the M component of the generated field is Z EM (t) = d3 xd4 x′ M ∗ (x)δ(z − z0 ) ×G(x; x′ )S(x′ ).

(18)

If the source is time-independent, then Equation (11) and the time-translation symmetry of G imply Z 1 EM (t) = d3 x′ M ∗ (x′ )S(x′ ). (19) βz Similarly, if a product E1 (x1 )E2 (x2 ) is given by a constant pair source S (2) (x) as Z E1 (x1 )E2 (x2 ) = d4 x′ G1 (x1 ; x′ )G2 (x2 ; x′ ) ×S (2) (x′ ).

(20)

then the time-integrated mode-projected component is Z Z 1 dt1 E1M1 (t1 )E2M2 (t2 ) = d3 x′ M1∗ (x′ ) β1z β2z ×M2∗ (x′ )S (2) (x′ ).

E.

(21)

We now introduce a χ(2) nonlinearity, which produces a nonlinear polarization that appears as a source term in the propagation equations. We consider three fields, “signal,” “idler” and “pump” with carrier frequencies ωs , ωi , ωp and wave-numbers ks , ki , kp , respectively. The respective field envelopes Es , Ei , Ep evolve according to Dp Ep = Ds Es = Di Ei =

First-order perturbation theory is sufficient to describe situations in which pairs are produced. For example, if E0s , E0i , E0p are source-free solutions, then Z 2 Es = E0s + ωs d4 x′ Gs (x; x′ ) † ×g(x′ )E0p (x′ )E0i (x′ ) + O(g 2 ).

(23)

and similar expressions for Ei , Ep are sufficient to give the lowest-order contribution to the pair-detection rate E D W (2) ∝ Es† Ei† Ei Es . Higher-order expansions would be necessary for double-pair production, etc.

F.

narrow-band frequency filters

In most down-conversion experiments, some sort of frequency filter is used. Assuming this filter is linear and stationary, the field reaching the detector is Z E (F ) (t) = dt′ F (t − t′ )E(t′ ) + G(t − t′ )Eres (t′ ). (24) Here Eres is a reservoir field required to maintain the field commutation relations. Assuming the reservoir is in the vacuum state, it will not produce E be D detections and can (F ) (F ) ignored. Defining HF (ti , ts ) ≡ Ei i (ti )Es s (ts ) , the fields that leave the filter obey Z HF (ti , ts ) = dt′ dt′′ Fi (ti − t′ )Fs (ts − t′′ ) × hEi (t′ )Es (t′′ )i .

Coupled wave equations

ωp2 gEs Ei exp[i∆kz] ωs2 gEp Ei† exp[−i∆kz] ωi2 gEp Es† exp[−i∆kz]

where g = −4m(x)d/c2 , d is the effective nonlinearity, equal to half the relevant projection of χ(2) , and ∆k ≡ kp − ks − ki is the wave-number mismatch. The dimensionless function m(x) describes the distribution of χ(2) . For example in a periodically-poled material it alternates between ±1. We can take ∆k = 0 without loss of generality, as the phase oscillation can be incorporated directly in the envelopes. The propagation directions (±) will be omitted unless needed for clarity. Note that for transparent materials χ(2) is real, and χ(2) (ωp ; ωs + ωi ) = χ(2) (ωs ; ωp − ωi ) = χ(2) (ωi ; ωp − ωs ).

(25)

In the narrowband case, i.e., when the correlation time between signal and idler is much less than the timescale of the impulse response functions, we can take ′ ′′ ′ ′′ hE R i (t )Es (t )i ≈ Aδ(t − t ) where the constant A ≡ dti hEi (ti )Es (ts )i. We find Z HF (ti , ts ) ≈ A dt′ Fi (ti − t′ )Fs (ts − t′ ) ≡ Af (ts − ti ).

(26)

With this, we see that the flux of pairs is (22)

W (2) (ts − ti ) =

4ns ni c2 ε20 |Af (ts − ti )|2 ~2 ω s ω i

(27)

5 with a total coincidence rate of W

(2)

Z

dti W (2) (ts − ti ) Z ns ni c2 ε20 2 = |A| dti |2f (ts − ti )|2 ~2 ω s ω i ns ni c2 ε20 |A|2 Γeff . ≡ ~2 ω s ω i =

A.

We consider first the process of SFG, for un-depleted signal and idler and no input pump. Signal and idler are constant and come from single-modes, (28)

We note that Γeff =

2 π

Z

dΩTs (Ω)Ti (−Ω)

4ωp2 d EMp (tp ) = −EMi EMs 2 c βz,p Z × d3 x′ Mp∗ (x′ )m(x′ )Mi (x′ )Ms (x′ ) ≡ −EMi EMs

(29)

R where Ts,i (Ω) ≡ | dt exp[iΩt]Fs,i (t)|2 are the signal and idler filter transmission spectra, respectively. For this reason we refer to Γeff as the effective line-width (in angular frequency) for the combined filters. Also important will be the singles rate D E (FS ) † (FS ) [E (t )] E (t ) W (1) = A−2 v s s s s γs gs Z 2ns cε0 = dt′ dt′′ Fs∗ (ts − t′ )Fs (ts − t′′ ) ~ωs

 × Es† (t′ )Es (t′′ ) Z 2ns cε0 C dt′ |Fs (ts − t′ )|2 ≈ ~ωs ns cε0 CΓeff,s (30) ≡ 2~ωs

 where C ≡ dt′ Es† (t′ )Es (t′′ ) . Γeff,s is the effective linewidth for the signal filter. R

IV.

sum-frequency generation

RESULTS

With the calculational tools described above, we now demonstrate the central results of this paper. We first express the efficiency of continuous-wave sum-frequency generation (SFG) in terms of a mode-overlap integral. This effectively reduces the non-linear optical problem to three uncoupled propagation problems. We then show that the efficiency of parametric down-conversion in the same medium is proportional to the SFG efficiency, for modes with the same shapes but opposite propagation direction. The constant of proportionality is found, allowing calculations of absolute efficiency based either on material properties such as χ(2) or measured SHG efficiencies. Similarly, the singles production efficiency is related to difference-frequency generation (DFG) and the collection efficiency is calculated. The same quantities for the degenerate case are also found.

4ωp2 d ISF G . c2 βz,p

(31)

The conversion efficiency is QSF G ≡ =

8ωp4 np d2 |ISF G |2 PMp = PMs PMi ns ni c5 ε0 |βz,p |2 2ωp2 d2 |ISF G |2 c3 ε 0 n p n s n i

(32)

The efficiency of a cw, single-mode source is thus proportional to the spatial overlap of the pump, signal, and idler modes, weighted by the nonlinear coupling g.

B.

non-degenerate parametric down-conversion

Next we consider the process of parametric downconversion. Using Equation (23), we can calculate to first order in g the correlation function Z hEi (xi )Es (xs )i = ωs2 d4 x′ Gs (xs , x′ ) D E † × E0,i (xi )E0,i (x′ ) ×g(x′ )E0,p (x′ ) Z ~ω 2 ω 2 = i 2i s d4 x′ Gs (xs , x′ ) c ε0 ×Gi (xi , x′ )g(x′ )E0,p (x′ )

(33)

For constant pump and single-mode collection we have Z dts hEMi (ti )EMs (ts )i AMi Ms ≡ =

≡

i~ωs ωi d EM c2 ε 0 n s n i p Z × d3 x′ Ms∗ (x′ )Mi∗ (x′ )m(x′ )Mp (x′ ) i~ωs ωi d EM IDC . c2 ε 0 n s n i p

(34)

∗ We note that IDC = ISF G . Also, the conjugate modes describe backward-propagating fields, as if the source

6 fields were sent through the nonlinear medium in the opposite direction. Thus if we want to know the brightness of down-conversion when all beams are propagating to the left, it is sufficient to calculate (or measure) the efficiency of up-conversion when all beams are propagating to the right. Using equations (32) and (34) we find |AMi Ms |2 =

C.

~2 ωi2 ωs2 Pp QSF G . 2 4c ε20 ns ni ωp2

(35)

generation. We find Z 2ωi2 d2 Pi = 3 Pp Ps d3 xi δ(zi − z0 ) c ε 0 ns ni np 2 Z × βz,i d4 x′ Gi (xi ; x′ )m(x′ )Mp (x′ )Ms∗ (x′ ) 2ωi2 d2 (s) |IDF G |2 0 ns ni np ≡ Pp Ps QDF G .

≡ Pp Ps

brightness

c3 ε

E.

We can now consider the brightness of the filtered, singlemode source. The rate of detection of pairs is ns ni c2 ε20 |A|2 Γeff ~2 ω s ω i ωi ωs Pp QSF G = Γeff 4ωp2

W (2) =

(36)

This simple expression is the first main result: The rate of pairs is simply the joint collection bandwidth Γeff , times the ratio of frequencies, times the pump power, times the up-conversion efficiency QSF G . Note that the last quantity can be calculated if the mode shapes and χ(2) (x) are known, for example in the paper of Boyd and Kleinman, or simulated for more complicated situations. Most importantly, it is directly measurable.

(38)

singles rates in PDC

We can find the rate of detection of singles in the mode MS by equation (30) and using Equation (A2) Z E D † ′ (x ) (x )E C = dts EM s M S s S Z = dts d3 xs d3 x′s Ms (xs )Ms∗ (x′s )

 ×δ(zs − z0 )δ(zs′ − z0 ) Es† (xs )Es (x′s ) Z |EMp |2 ωs4 = d3 xd3 x′ Ms (x)g(x)Mp∗ (x) |βz,s |2 D E † × E0i (x)E0i (x′ ) Ms∗ (x′ )g(x′ )Mp (x′ ) Z 2~ωi ω 2 d2 = 3 2 s |EMp |2 d4 x′′ δ(z ′′ − z0 ) c ns ni ε 0 2 Z 3 ′′ ∗ × βi,z d xGi (x ; x)Ms (x)m(x)Mp (x) (39) so that

D.

We now consider the classical situation in which pump and signal beam are injected into the crystal and idler is generated. We will see that this directly measurable process is related to the singles generation rate by parametric down-conversion. The generated idler is Ei (x) =

ωi2

Z

If pump and signal are from modes MP , MS , respectively, we find Z

F.

(40)

conditional efficiency

The conditional efficiency for the idler (probability of collecting the idler, given that the signal was collected) is

∗ d4 x′ Gi (x; x′ )g(x′ )E0p (x′ )E0s (x′ ).

4ω 2 d ∗ Ei (x) = − 2i EMp (tp )EM (ts ) s c ×m(x′ )Mp (x′ )Ms∗ (x′ ).

cωi ωs (s) Pp |IDF G |2 Γeff,s 32np ns ni ε0 ωs (s) = Γeff,s Pp QDF G 4ωi

W (1) =

difference-frequency generation

ηs ≡

W (2) (1)

Ws

=

Γeff |ISF G |2 Γs |I (s) |2 DF G

(41)

d4 x′ Gi (x; x′ ) (37)

R The total power generated is Pi = 2cni ε0 d3 xi δ(zi − z0 )|Ei (xi )|2 where z0 indicates a plane downstream of the

G.

degenerate processes

Up to this point, we have discussed only non-degenerate processes, i.e., those in which the signal and idler fields

7 are distinct and do not interfere. This is always the case for type-II down-conversion, and will be the case for typeI down-conversion if the frequencies and/or directions of propagation are significantly different. We now consider degenerate processes, in which there is only one downconverted field (signal). The above discussion is modified only slightly. The signal and pump evolve by 1 2 ω gEs Es 2 p = ωs2 gEp Es† .

Dp Ep = Ds Es

(42)

term is the phase-independent contribution to the gain, an experimentally accessible quantity. We have

δP ≡ hPs − P0,s iφs Z Z = 2ns cε0 d3 xs δ(zs − z0 )|E1,s (xs )|2 Z 8ωs2 ε0 d2 = |E0s |2 |Ep |2 d4 xs δ(zs − z0 ) cn s Z × βz,s d3 x′ Gs (xs ; x′ ) 2

× m(x′ )Mp (x′ )Ms∗ (x′ )|

H.

2ωs2 d2 |IAP G |2 c3 n2s np ε0 ≡ P0s Pp QAP G .

≡ P0s Pp

second harmonic generation

(48)

The calculation of second-harmonic generation (SHG) proceeds exactly as in sum-frequency generation, except for the factor of one half and with all “idler” variables replaced by “signal” variables. Thus we find Pp = Ps2 QSHG

(43) J.

where QSHG =

ωp2 d2 |ISHG |2 3 2c ε0 np n2s

(44)

and ISHG ≡

I.

Z

d3 xMp∗ (x)m(x)Ms (x)Ms (x).

Next we consider the process of degenerate parametric down-conversion, for which

(45) Es = E0s + ωs2

Z

d4 x′ Gs (x; x′ )

† ×g(x′ )E0p (x′ )E0s (x′ ).

average parametric gain

The other classical process of interest is parametric amplification of the signal by the pump. The first-order solution for the signal field is Z Es = E0s + ωs2 d4 x′ Gs (x; x′ ) † (x′ ) ×g(x′ )E0p (x′ )E0s ≡ E0s + E1s .

degenerate PDC

We find the correlation function

hEs (xs )Es (x′s )i


∗ +2Re[E0,s (xs )E1,s (xs )] + |E1,s (xs )|2 . (47)

The first term is the input signal power P0,s , the second term depends on the relative phase φp − 2φs , and the last

=

ωs2

Z

d4 x′′ Gs (xs , x′′ ) D E † × E0,s (x′s )E0,s (x′′ ) g(x′′ )

×E0,p (x′′ )

(46)

The signal power at the output is Z Ps = 2ns cε0 d3 xs δ(zs − z0 )|Es (xs )|2 Z = 2ns cε0 d3 xs δ(zs − z0 ) |E0,s (xs )|2

(49)

(50)

at which point it is clear that the only difference from the non-degenerate case of Eq. (33)will be the replacement of idler variables with signal variables. We find

W (2) = Γeff

Γeff ωs2 Pp QSHG = Pp QSHG . 4ωp2 16

(51)

8 K.

Singles rates (degenerate)

As before, we can find the rate of detection of singles in the mode MS by equation (30) and Z D E † ′′ ′ C = dt′s EM (t ) (t )E M S s s S Z = dt′s d3 x′s d3 x′s Ms (x′s )Ms∗ (x′′s )

 ×δ(zs′ − z0 )δ(zs′′ − z0 ) Es† (x′s )Es (x′′s ) Z |EMp |2 ωs4 = d3 x′ d3 x′′ Ms (x′ )g(x′ )Mp∗ (x′ ) |βz,s |2 D E † × E0i (x′ )E0i (x′′ ) Ms∗ (x′′ )g(x′′ )Mp (x′′ ) Z 2~ω 3 d2 = 3 3s |EMp |2 d4 x′′ δ(z ′′ − z0 ) c ns ε 0 2 Z 3 ′′ ∗ × βs,z d xGs (x ; x)Ms (x)m(x)Mp (x) (. 52)

where m ∈ {s, i, p}, r is the radial component of x, and q ≡ z − izR where zR is the Rayleigh range, assumed equal for all beams. We assume a periodically-poled material in which χ(2) (z) alternates with period 2π/Q so and we approximate m(x) ≈ exp[iQz]deff /d. From Equation (31) we find r −i∆kz ′ 3 Z L/2 kp ks ki zR ′e ISF G = dz π3 q|q|2 −L/2 Z ” ′2 “k k +k −i q∗p − s q i r2 × 2πr′ dr′ e r 3 2i kp ks ki zR = k+ π Z L/2 ′ e−i∆kz (56) × dz ′ ′ (z − izR ) (Rk z ′ + izR ) −L/2 where ∆k ≡ kp − ks − ki − Q and Rk ≡ k− /k+ and k± ≡ kp ± (ks + ki ). In terms of the dimensionless variables κ ≡ ∆kL, ζ ≡ z ′ /L, ζR ≡ zR /L we find

The singles rate is thus Ws(1)

1 (s) = Γeff,s Pp QAP G . 4

(53)

ISF G = where Υ ≡

L.

Conditional efficiency (degenerate)

ηs ≡

(1) Ws

Γeff |ISHG |2 = Γeff,s |I (s) |2

(54)

QSF G =

1/2

dζ

−1/2

e−iκζ . (ζ − iζR ) (Rk ζ + iζR )

(58)

8πωp2 kp ks ki zR d2eff |Υ|2 . 2 3 c ε0 np ns ni k+

(59)

AP G

A.

V.

Z

(57)

From Equation (32) the upconversion efficiency is then

The conditional efficiency is W (2)

ζR 2π

2i p πkp ks ki zR Υ k+

EXAMPLE CALCULATIONS

We now illustrate the preceding, general results with a few special cases. We first calculate the overlap integral for co-propagating gaussian beams. This allows us to 1) compare our results to the classical results of Boyd and Kleinman [29], 2) predict absolute brightness for an important geometry, type-II co-linear down-conversion in quasi-phase-matched material. Also, we compare to a recent calculation of absolute brightness for a specific geometry by Ling, et al. [17]. We consider collinear, frequency-degenerate type-II PDC with circular gaussian beams for signal, idler and pump. We take mode shape functions r km zR 1 ikm z ikm r2q2 . (55) Mm (x) = e e π q

Boyd and Klienman, 1968

With this expression we can compare our results to those of Boyd and Kleinman [29] for the case of secondharmonic generation. As that calculation does not include quasi-phase matching, we take Q = 0, and then for any reasonable phase-matching we have np ≈ ns , kp ≈ 2ks and thus k− = Rk = 0, k+ ≈ 2kp . We note that for Rk = 0, Υ becomes equal to the function H of Boyd and Kleinman for zero absorption and walk-off angle. We find Pp =

4πks ωs2 zR d2 |Υ|2 Ps2 . c3 ε0 n2s np

(60)

Boyd and Kleinman find in Equations (2.16),(2.17) and (2.20) to (2.24) P2 =

128π 2 ω12 2 2 2π 2 zR d P Lk |H|2 1 1 2 c3 n 1 n 2 L

(61)

9 or P2 =

256π 4 k1 ω12 zR d2 |H|2 P12 . c3 n21 n2

(62)

When converting this expression to MKS units, d2 → d2 /64π 3 ε0 , and we see that the two calculations agree.

B.

Type-II collinear brightness

The integral Υ must be evaluated numerically. For a 1 cm crystal of PPKTP and a vacuum wavelength λs = λi = 800 nm we have (ns , ni , np ) = (1.844, 1.757, 1.964) and deff = 2.4 pm/V so that Rk = 0.04 and the maximum of (zR /L)|Υ|2 ≈ 0.054 occurs at κ ≈ −3.0, ζR ≈ 0.18. We find QSF G = 2.0 × 10−3 W−1 . Used as a photon-pair source, this same crystal and geometry would yield by Equation (36) QSF G 16

(63)

or a pair generation efficiency of QSF G /16 = 0.8 pairs (s mW MHz)−1 . Note that Γeff is the filter bandwidth in angular frequency.

C.

Ling, Lamas-Linares, and Kurtsiefer, 2008

Recently, Ling et al. calculated the absolute emission rate into gaussian modes in the thin-crystal limit of (nonperiodically-poled) nonlinear material [17]. They arrive to a down-conversion spectral brightness of dR(ωs ) = dωs

deff αs αi Ep0 Φ(∆k) c

!2

ωs ωi 2πns ni

where R is the pair collection rate and Z Z Φ(∆k) ≡ dz dy dx ei∆k·r Up (r)Us (r)Ui (r).

ωs ωi d2 Pp |ISF G |2 Γeff . 3 2c ε0 np ns ni

(68)

Then with Equation (29) and noting that in the thin crystal limit |ISF G | = |αp αs αi Φ(∆k)|, we see that the two results are identical.

VI.

CONCLUSIONS

Using the approach of coupled wave-equations, familiar from nonlinear optics, we have calculated the absolute brightness and temporal correlations of spontaneous parametric down-conversion in the narrow-band regime. The results are obtained with a Green function method and are generally valid within the paraxial regime. We find that efficiencies of SFG and SPDC can be expressed in terms of mode overlap integrals, and are proportional for corresponding geometries. Also, we find pair time correlations in terms of signal and idler filter impulse response functions. Results for both degenerate and nondegenerate SPDC are found. Comparison to classical calculations by Boyd and Kleinman, and to a recent calculation by Ling et al. show the connection to classical nonlinear optics and “golden rule”-style brightness calculations, while considerably generalizing the latter. We expect these results to be important both for designing SPDC sources, as the results of well-known classical calculations can be used, and for building and optimizing such sources.

(64) APPENDIX A: ALTERNATE PROPAGATOR

(65)

Here Um describe the mode shapes p of the form Um (r) = 2 ikm z −(x2 +y 2 )/Wm 2 are normal2/πWm e e and αm = 0 ization constants. The field Ep is defined such that |Ep0 |2 = 2α2p Pp /ε0 np c where Pp is the pump power, giving ωs ωi d2 dR(ωs ) = 3 Pp |αp αs αi Φ(∆k)|2 dωs πc ε0 np ns ni

where we have assumed dR(ωs )/dωs constant over the width of the filters. For comparison, using Eqs (32) and (36), we find W (2) =

Next we make a numerical calculation for frequencydegenerate type-II SPDC, a geometry of current interest for generation of entangled pairs, for example.

W (2) = Γeff Pp

Assuming the output is collected with narrow-band filters of transmission Ts (ωs ), Ti (ωi ) for signal and idler, respectively, the integrated rate is Z dR(ωs ) R= dΩTs (ωp /2 + Ω)Ti (ωp /2 − Ω) (67) dωs

(66)

We can use Equations (12) and (13) to express the propagator as Z

 † ′ 2 E(x)E (x ) = |βt | d4 x′′ d4 x′′′ δ(t′′ − tf )

 ×δ(t′′′ − tf ) E(x′′ )E † (x′′′ ) ×G ∗ (x′′ ; x)G(x′′′ ; x′ ). Z 2 = |Aγ βt | d4 x′′ δ(t′′ − tf ) ×G ∗ (x′′ ; x)G(x′′ ; x′ )

(A1)

10 R 4 ′′ ′′ Noting d x δ(t − tf )G ∗ (x′′ ; x)G(x′′ ; x′ ) R 4 ′′that′′ vg d x δ(z − z0 )G ∗ (x′′ ; x)G(x′′ ; x′ ) we find

 2~nω 3 E(x)E † (x′ ) = d4 x′′ δ(z ′′ − z0 ) c3 ε 0 ×G ∗ (x′′ ; x)G(x′′ ; x′ ) Z

=

Γi f (ts − ti ) = 2

Γ θ(τ ) exp[−Γτ /2]. 2 2

2

or 

exp[−Γs τ /2] τ > 0 exp[Γi τ /2] τ < 0

The effective bandwidth is Z Γeff = 4 dτ |f (τ )|2 =

(B5)

f (ts − ti ) =

Γ exp[−Γ|ts − ti |/2]. 4

(B6)

(B1)

The spectral transmission is T (Ω) = Γ /(Γ + 4Ω ), i.e., unit transmission for constant E, a full-width at halfR maximum of ∆ΩFWHM = Γ and an area dΩ T (Ω) = πΓ/2. If we put a filter of this sort in each arm, the output has Z Γs Γi dt′ θ(ts − t′ )θ(ti − t′ ) f (ts − ti ) = 4 × exp[−Γi (ti − t′ )/2] × exp[−Γs (ts − t′ )/2] (B2)

Γs Γi f (τ ) = 2(Γs + Γi )

0 t i < ts exp[−Γi (ti − ts )/2] ti > ts

That is, the idler photon will always arrive later, and with a distribution (after the signal arrival) that is precisely the transfer function of the idler-beam filter. Another interesting limit is for matched filters, Γs = Γi = Γ. Then we find

A common filter has a Lorentzian transfer function and an exponential impulse response

2



(A2)

APPENDIX B: LORENTZIAN FILTER

F (τ ) =

terms, not having a filter there at all), filter becomes

Note that for Γs → ∞, the detection rate is |A|2 Γi /4, i.e., proportional to the idler filter bandwidth Γi . The reverse, s ↔ i is also true, of course. From this we can get an idea of the conditional efficiency: The rate for filtered signal with any idler is proportional to Γs ≥

Γi Γs . Γs + Γi

(B7)

For example, putting matched filters Γs = Γi = Γ will give a rate proportional to Γi Γs /(Γs + Γi ) i.e., half of the rate without the idler filter. This indicates that, of the signal photons that pass the the signal filter, half of their “twin” idler photons do not pass the idler filter.

(B3) Acknowledgements

Γs Γi Γs + Γi

(B4)

It is worth noting that in the limit Γs → ∞ (the limit of a broad-band filter in the signal beam, or in practical

We thank A. Cer`e, F. Wolfgramm, G. Molina, A. Haase, and N. Piro for helpful discussions. This work was supported by the Spanish MEC under the ILUMA project (Ref. FIS2008-01051), the Consolider-Ingenio 2010 Project “QOIT” and by Marie Curie RTN “EMALI.”

[1] M. Fejer, G. Magel, D. Jundt, and R. Byer, IEEE JOURNAL OF QUANTUM ELECTRONICS 28, 2631 (1992). [2] P. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. Sergienko, and Y. Shih, PHYSICAL REVIEW LETTERS 75, 4337 (1995). [3] P. Kwiat, E. Waks, A. White, I. Appelbaum, and P. Eberhard, PHYSICAL REVIEW A 60, R773 (1999). [4] G. Giorgi, G. Di nepi, P. Mataloni, and F. De martini, LASER PHYSICS 13, 350 (2003). [5] M. Fiorentino, G. Messin, C. Kuklewicz, F. Wong, and J. Shapiro, PHYSICAL REVIEW A 69, 041801 (2004). [6] M. Pelton, P. Marsden, D. Ljunggren, M. Tengner, A. Karlsson, A. Fragemann, C. Canalias, and F. Laurell, OPTICS EXPRESS 12, 3573 (2004).

[7] C. Kuklewicz, M. Fiorentino, G. Messin, F. Wong, and J. Shapiro, PHYSICAL REVIEW A 69, 013807 (2004). [8] M. Fiorentino, C. Kuklewicz, and F. Wong, OPTICS EXPRESS 13, 127 (2005). [9] F. Wolfgramm, X. Xing, A. Cere, A. Predojevic, A. Steinberg, and M. Mitchell, OPTICS EXPRESS 16, 18145 (2008). [10] J. O’brien, G. Pryde, A. White, T. Ralph, and D. Branning, Nature 426, 264 (2003). [11] S. Groblacher, T. Paterek, R. Kaltenbaek, C. Brukner, M. Zukowski, M. Aspelmeyer, and A. Zeilinger, Nature 446, 871 (2007). [12] B. Higgins, D. Berry, S. Bartlett, H. Wiseman, and G. Pryde, Nature 450, 393 (2007).

11 [13] S. Sauge, M. Swillo, S. Albert-Seifried, G. Xavier, J. Waldeback, M. Tengner, D. Ljunggren, and A. Karlsson, OPTICS EXPRESS 15, 6926 (2007). [14] A. Haase, N. Piro, J. Eschner, and M. W. Mitchell, OPTICS LETTERS 34, 5557 (2009). [15] D. Ljunggren and M. Tengner, PHYSICAL REVIEW A 72, 062301 (2005). [16] C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, PHYSICAL REVIEW A 64, art. no. (2001). [17] A. Ling, A. Lamas-Linares, and C. Kurtsiefer, PHYSICAL REVIEW A 77, 043834 (2008). [18] D. Kleinman, PHYSICAL REVIEW 174, 1027 (1968). [19] Y. Zeldovich and D. Klyshko, JETP LETTERS-USSR 9, 40 (1969). [20] Y. R. Shen, The Principles of nonlinear optics (WileyInterscience, New York, 1984), 1st ed. [21] D. Klyshko, ZHURNAL EKSPERIMENTALNOI I TEORETICHESKOI FIZIKI 94, 82 (1988). [22] D. Burnham and D. Weinberg, PHYSICAL REVIEW LETTERS 25, 84 (1970).

[23] B. Mollow, PHYSICAL REVIEW A 8, 2684 (1973). [24] C. Hong and L. Mandel, PHYSICAL REVIEW A 31, 2409 (1985). [25] R. Ghosh, C. Hong, Z. Ou, and L. Mandel, PHYSICAL REVIEW A 34, 3962 (1986). [26] A. Valencia, A. Cere, X. Shi, G. Molina-Terriza, and J. Torres, PHYSICAL REVIEW LETTERS 99 (2007). [27] P. Mosley, J. Lundeen, B. Smith, P. Wasylczyk, A. U’ren, C. Silberhorn, and I. Walmsley, PHYSICAL REVIEW LETTERS 100, 133601 (2008). [28] R. W. Boyd, Nonlinear Optics (Academic Press, Burlington, MA, 2008), 3rd ed. [29] G. Boyd and D. Kleinman, JOURNAL OF APPLIED PHYSICS 39, 3597 (1968). [30] We note that Hamiltonian-based treatments often do not about the constant preceding the integral R 3 agree d xχ(2) E 3 , not even its sign. This question does not present a problem for the present calculation, which does not employ a Hamiltonian.